Finite Math Examples

$f (x) = - 0.4 x^{2} + 2 x + 5$

Step 1

Set $- 0.4 x^{2} + 2 x + 5$ equal to $0$ .

$- 0.4 x^{2} + 2 x + 5 = 0$

Step 2

Solve for $x$ .

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Step 2.1

Factor $- 0.2$ out of $- 0.4 x^{2} + 2 x + 5$ .

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Step 2.1.1

Factor $- 0.2$ out of $- 0.4 x^{2}$ .

$- 0.2 (2 x^{2}) + 2 x + 5 = 0$

Step 2.1.2

Factor $- 0.2$ out of $2 x$ .

$- 0.2 (2 x^{2}) - 0.2 (- 10 x) + 5 = 0$

Step 2.1.3

Factor $- 0.2$ out of $5$ .

$- 0.2 (2 x^{2}) - 0.2 (- 10 x) - 0.2 \cdot - 25 = 0$

Step 2.1.4

Factor $- 0.2$ out of $- 0.2 (2 x^{2}) - 0.2 (- 10 x)$ .

$- 0.2 (2 x^{2} - 10 x) - 0.2 \cdot - 25 = 0$

Step 2.1.5

Factor $- 0.2$ out of $- 0.2 (2 x^{2} - 10 x) - 0.2 (- 25)$ .

$- 0.2 (2 x^{2} - 10 x - 25) = 0$

Step 2.2

Divide each term in $- 0.2 (2 x^{2} - 10 x - 25) = 0$ by $- 0.2$ and simplify.

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Step 2.2.1

Divide each term in $- 0.2 (2 x^{2} - 10 x - 25) = 0$ by $- 0.2$ .

$\frac{- 0.2 (2 x^{2} - 10 x - 25)}{- 0.2} = \frac{0}{- 0.2}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor of $- 0.2$ .

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Step 2.2.2.1.1

Cancel the common factor.

$\frac{- 0.2 (2 x^{2} - 10 x - 25)}{- 0.2} = \frac{0}{- 0.2}$

Step 2.2.2.1.2

Divide $2 x^{2} - 10 x - 25$ by $1$ .

$2 x^{2} - 10 x - 25 = \frac{0}{- 0.2}$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Divide $0$ by $- 0.2$ .

$2 x^{2} - 10 x - 25 = 0$

Step 2.3

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 2.4

Substitute the values $a = 2$ , $b = - 10$ , and $c = - 25$ into the quadratic formula and solve for $x$ .

$\frac{10 \pm \sqrt{{(- 10)}^{2} - 4 \cdot (2 \cdot - 25)}}{2 \cdot 2}$

Step 2.5

Simplify.

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Step 2.5.1

Simplify the numerator.

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Step 2.5.1.1

Raise $- 10$ to the power of $2$ .

$x = \frac{10 \pm \sqrt{100 - 4 \cdot 2 \cdot - 25}}{2 \cdot 2}$

Step 2.5.1.2

Multiply $- 4 \cdot 2 \cdot - 25$ .

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Step 2.5.1.2.1

Multiply $- 4$ by $2$ .

$x = \frac{10 \pm \sqrt{100 - 8 \cdot - 25}}{2 \cdot 2}$

Step 2.5.1.2.2

Multiply $- 8$ by $- 25$ .

$x = \frac{10 \pm \sqrt{100 + 200}}{2 \cdot 2}$

Step 2.5.1.3

Add $100$ and $200$ .

$x = \frac{10 \pm \sqrt{300}}{2 \cdot 2}$

Step 2.5.1.4

Rewrite $300$ as $10^{2} \cdot 3$ .

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Step 2.5.1.4.1

Factor $100$ out of $300$ .

$x = \frac{10 \pm \sqrt{100 (3)}}{2 \cdot 2}$

Step 2.5.1.4.2

Rewrite $100$ as $10^{2}$ .

$x = \frac{10 \pm \sqrt{10^{2} \cdot 3}}{2 \cdot 2}$

Step 2.5.1.5

Pull terms out from under the radical.

$x = \frac{10 \pm 10 \sqrt{3}}{2 \cdot 2}$

Step 2.5.2

Multiply $2$ by $2$ .

$x = \frac{10 \pm 10 \sqrt{3}}{4}$

Step 2.5.3

Simplify $\frac{10 \pm 10 \sqrt{3}}{4}$ .

$x = \frac{5 \pm 5 \sqrt{3}}{2}$

Step 2.6

The final answer is the combination of both solutions.

$x = \frac{5 + 5 \sqrt{3}}{2}, \frac{5 - 5 \sqrt{3}}{2}$

$x = \frac{5 \pm 5 \sqrt{3}}{2}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$x = \frac{5 \pm 5 \sqrt{3}}{2}$

Decimal Form:

$x = 6.83012701 \dots, - 1.83012701 \dots$

Step 4